In this section we will construct the cube complex \(\mathfrak {X}_{d,r}(O,Y)\) . We remark that our complex is similar in spirit to the one introduced by Genevois–Lonjou–Urech in [1]} for establishing the finiteness properties of a related family of related groups, and is inspired by the Stein–Farley complexes associated to Higman–Thompson groups [2]}, [3]}.
